In recent years, multi-branched polymers (star-shaped polymers) have drawn attention because of distinctive properties thereof, such as low viscosity, particulate properties, and molecular functions. Accordingly, various techniques for synthesis of such polymers have been known. The technique for synthesis disclosed in JP 2010-506002 A (Patent Document 1) involves the use of, as a catalyst, a lanthanoid compound, a nickel-containing compound, or an organic metal compound. Thus, a process of purification thereof is necessary. JP 2003-252919 A (Patent Document 2) describes a polyfunctional radical polymerization initiator, which is preferably used for the production of a multi-branched polymer, composed of a compound comprising 3 or more ethylenic double bonds in a single molecule and a boron compound added to the double bonds (see the abstract).